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Summary

With many new concrete examples and historical notes, Topological Vector Spaces, Second Edition provides one of the most thorough and up-to-date treatments of the Hahn–Banach theorem. This edition explores the theorem’s connection with the axiom of choice, discusses the uniqueness of Hahn–Banach extensions, and includes an entirely new chapter on vector-valued Hahn–Banach theorems. It also considers different approaches to the Banach–Stone theorem as well as variations of the theorem.

The book covers locally convex spaces; barreled, bornological, and webbed spaces; and reflexivity. It traces the development of various theorems from their earliest beginnings to present day, providing historical notes to place the results in context. The authors also chronicle the lives of key mathematicians, including Stefan Banach and Eduard Helly.

Suitable for both beginners and experienced researchers, this book contains an abundance of examples, exercises of varying levels of difficulty with many hints, and an extensive bibliography and index.

Hahn–Banach TheoremsWhat Is It?The Obvious SolutionDominated and Continuous ExtensionsConsequencesThe Mazur–Orlicz TheoremMinimal Sublinear FunctionalsGeometric Form Separation of Convex SetsOrigin of the TheoremFunctional Problem SolvedThe Axiom of ChoiceNotes on the Hahn–Banach TheoremHelly

DualityPaired SpacesWeak TopologiesPolarsAlaogluPolar TopologiesEquicontinuityTopologies of PairsPermanence in DualityOrthogonalsAdjointsAdjoints and ContinuitySubspaces and QuotientsOpenness of Linear MapsLocal Convexity and HBEP

Author(s) Bio

Lawrence Narici and Edward Beckenstein are professors of mathematics at St. John’s University in New York.

Reviews

Besides a general renovation, the text has improved the topics related to the Hahn-Banach theorem … there is a whole new chapter on vector-valued Hahn-Banach theorems and an enlarged presentation of the Banach-Stone theorems. The text remains a nice expository book on the fundamentals of the theory of topological vector spaces. —Luis Manuel Sanchez Ruiz, Mathematical Reviews, Issue 2012a

This is a nicely written, easy-to-read expository book of the classical theory of topological vector spaces. … The proofs are complete and very detailed. … The comprehensive exposition and the quantity and variety of exercises make the book really useful for beginners and make the material more easily accessible than the excellent classical monographs by Köthe or Schaefer. … this is a well-written book, with comprehensive proofs, many exercises and informative new sections of historical character, that presents in an accessible way the classical theory of locally convex topological vector spaces and that can be useful especially for beginners interested in this topic.—José Bonet, Zentralblatt MATH 1219

Praise for the First Edition:This is a very carefully written introduction to topological vector spaces. But it is more. The enthusiasm of the authors for their subject, their untiring efforts to motivate and explain the ideas and proofs, and the abundance of well-chosen exercises make the book an initiation into a fascinating new world. The reader will feel that he does not get only one aspect of this field but that he really gets the whole picture.—Gottfried Köthe, Rendiconti del Circolo Matematico di Palermo, Series II, Volume 35, Number 3, September 1986

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